We think you are located in South Africa. Is this correct?

Summary

Don't get left behind

Join thousands of learners improving their maths marks online with Siyavula Practice.

Sign up here

5.6 Summary (EMCGY)

Terminology:
ExpressionA term or group of terms consisting of numbers, variables and the basic operators (\(+, -, \times, \div\)).
Univariate expressionAn expression containing only one variable.
Root/ZeroA root, also referred to as the “zero”, of an equation is the value of \(x\) such that \(f(x)=0\) is satisfied.
Polynomial

An expression that involves one or more variables having different powers and coefficients.

\(a_{n}x^{n} + \ldots + a_2x^{2} + a_{1}x + a_{0}, \text{ where } n \in \mathbb{N}_0\)

Monomial

A polynomial with one term.

For example, \(7a^{2}b \text{ or } 15xyz^{2}\).

Binomial

A polynomial that has two terms.

For example, \(2x + 5z \text{ or } 26 - g^{2}k\).

Trinomial

A polynomial that has three terms.

For example, \(a - b + c \text{ or } 4x^2 + 17xy - y^3\).

Degree/Order

The degree, also called the order, of a univariate polynomial is the value of the highest exponent in the polynomial.

For example, \(7p - 12p^2 + 3p^5 + 8\) has a degree of \(\text{5}\).

  • Quadratic formula: \(x = \frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\)

  • Remainder theorem: a polynomial \(p(x)\) divided by \(cx - d\) gives a remainder of \(p\left(\dfrac{d}{c}\right)\).

  • Factor theorem: if the polynomial \(p(x)\) is divided by \(cx - d\) and the remainder, \(p \left( \frac{d}{c} \right)\), is equal to zero, then \(cx - d\) is a factor of \(p(x)\).

  • Converse of the factor theorem: if \(cx - d\) is a factor of \(p(x)\), then \(p \left( \frac{d}{c} \right) = 0\).

  • Synthetic division:

    90dce9977541474c65c21a1e6c2ca835.png

    We determine the coefficients of the quotient by calculating:

    \begin{align*} q_{2} &= a_{3} + \left( q_{3} \times \frac{d}{c} \right) \\ &= a_{3} \quad \text{ (since } q_{3} = 0) \\ q_{1} &= a_{2} + \left( q_{2} \times \frac{d}{c} \right) \\ q_{0} &= a_{1} + \left( q_{1} \times \frac{d}{c} \right) \\ R &= a_{0} + \left( q_{0} \times \frac{d}{c} \right) \end{align*}